Until recently, the even Coulomb potential \(-\frac{\alpha }{|x|}\)  was considered impenetrable; the Hamiltonian of 1D hydrogen atom was assumed to be the direct sum \(\mathcal {D}_-\oplus \mathcal {D}_+\)  of the half-line Schrödinger operators
\(\mathcal {D}_\pm =-\frac{d^2}{dx^2}-\frac{\alpha }{|x|}\)  on \(\mathbb {R}_\pm \)  with the Dirichlet boundary condition at zero. Many authors have obtained this result [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]} approximating the Coulomb potential in different ways, e.g.,
\(-\alpha \min \left\lbrace \dfrac{1}{|x|},\dfrac{1}{\varepsilon }\right\rbrace ,\quad -\dfrac{\alpha }{|x|+\varepsilon }, \quad -\frac{\alpha }{\sqrt{x^2+\varepsilon ^2}},\quad -\frac{\alpha |x|}{x^2+\varepsilon ^2}.\) 
